SlideShare a Scribd company logo
1 of 13
Download to read offline
Mathematics in Middle and Upper Primary
                                                                               Fall
                                                                          EDUC8505     08
Rationale

Mathematics is a discipline which has evolved from the human need to measure and
communicate about time, quantity and space (Moursund, 2002).           It is inherently
abstract, applicable over a wide field and uses symbols to represent mathematical
concepts.

Traditional theoretical frameworks associated with children’s mathematical thinking
include empiricism, where knowledge is external and acquired through the senses,
(neo)nativism/rationalism which emphasises the in-born capabilities of the child to
reason,     and interactionalism, which recognises interacting roles of nature and
experience, and considers the child as active in knowledge construction (Lester, 2007).

A central part of each of these frameworks is experiences, which allow children to
internalise or express knowledge. Experiences provide opportunities to learn, which
are considered “the single most important predictor of student achievement” (National
Research Council, 2001, p334; cited in Lester, 2007), and allow children to acquire
physical, socio-conventional and logico-mathematical knowledge (Piaget, 1967, cited
in Kamii, 2004). They are instrumental in supporting student affect, which plays a
crucial role in mathematics teaching and learning (Hart&Walker, 1993, cited in
Baroody, 1998).

Experiences should illustrate a wide variety of examples relating to the key concept in
different contexts, to facilitate students forming multiple representations and
connections, and building conceptual understanding, rather than simply applying
procedural knowledge.      Brownell (cited in Lester, 2007) refers to conceptual
understanding as mental connections among mathematical facts, procedures and
ideas.    Vergnaud (1983, cited in Lester, 2007) introduced the concept of the
multiplicative conceptual field, a complex system of interrelated concepts, student
ideas (competencies and misconceptions), procedures, problems, representations,
objects, properties and relationships that cannot be studied in isolation, including
multiplication, division, fractions, ratios, simple and multiple proportions, rational
numbers, dimensional analysis and vector spaces. The programme provides varied
activities representing the five sub-constructs of fractions (Kieren, 1980, cited in Way

Sharon McCleary
             5
Mathematics in Middle and Upper Primary                                   EDUC8505


& Bobis, 2011) part-whole, measure, quotient, operator, ratio), and encourages
connections between different rational numbers (e.g. Lesson 6: Fractions as quotients,
incorporating relational concepts and using numbers with common factors, which
support richer interconnections (Empson, 2005)).




Planned experiences should also aim to expose misconceptions, prevent the formation
of new ones (Bottle, 2005), and cater to students of different ability levels by using
open-ended activities. The programme achieves this by selecting activities that target
common misconceptions and require students to disprove them with concrete
materials (e.g. Lesson 3: Show Me A Half).

Concrete materials are central to assisting students in the concrete operations phase of
development understand mathematical concepts (Kamii, 2004). However their use
does not automatically result in mathematical learning, as students can focus on
unintended aspects and fail to abstract the intended concept             (Gray, 1999).
Consequently, opportunities for exploring the manipulative before using it are given
(Lesson 4: Exploring Pattern Blocks), and use is closely aligned with conceptual
understanding and linked to symbolic conventions in order to promote purposeful
connections in students’ minds.

Several frameworks characterise mathematical learning as progressing from
physical/concrete   interaction,   to   generalising   abstract   ideas/concepts    and
representing them symbolically (Cowan, 2006; Lester, 2007; Baroody, 1998). Visual
imagery constructed from concrete experiences is central to this progression, and its
role in assisting learning has been addressed by several researchers, including
Presmeg, Goldin and Thomas. Consequently, several experiences in the programme
use concrete materials to encourage clear visual images that may assist children in
thinking mathematically (e.g. Lesson 1: water in glasses, Lessons 4&5: Pattern Blocks).

Correct mathematical language and writing conventions are also crucial to this
process, since effective communication is pivotal in clarifying inconsistencies between
the child’s inner understandings and correct conceptual understanding, and in
allowing opportunities for exchanges between peers, expanding strategy knowledge
through social learning (Vygotsky, 1978). The teacher’s role is to provide clear links

Sharon McCleary                                                                       2
Mathematics in Middle and Upper Primary                                 EDUC8505


between concepts and conventional language/symbols, enabling semiotic meaning
making without stifling inherent thought processes. Opportunities to build fluency are
provided in each lesson of the programme through reading, writing, talking and
listening.

Another feature linked to developing conceptual understanding is allowing students to
actively expend effort in making sense of important mathematical ideas. Festinger’s
(1957) theory of cognitive dissonance describes perplexity as a central impetus for
cognitive growth, and Hatano (1988) identifies cognitive incongruity as the critical
trigger for developing reasoning skills that display conceptual understanding (Lester,
2007). This is consistent with constructivist ideas of presenting problems near the
boundary of the student’s Zone of Proximal Development (Vygotsky, 1978), allowing
sufficient challenge to promote thinking and application of conceptual knowledge,
while supporting opportunities for success and maintaining positive affect: “Acquired
knowledge is most useful to a learner when it is discovered through their own
cognitive efforts, related to and used in reference to what one has known before”
(Bruner, cited in Cowan, 2006, pg 26). The programme incorporates problem solving
allowing different solution methods: Lessons 6 & 12 provide additive and
multiplicative thinking arising from invented strategies for division questions,
allowing students opportunities to ‘struggle’ with relevant mathematical concepts in
authentic scenarios.

The teacher’s role encompasses providing children with engaging, challenging and
enjoyable experiences which emphasise conceptual understanding and promote a
positive attitude towards mathematics.     Implicit in this is creating a classroom
environment which allows opportunities for discussion, assists students in becoming
fluent with conventional mathematical language/symbols and is accepting of invented
strategies and solution methods.     This facilitates students’ forming connections
between multiple representations and abstracting meaning from experiences to
progress and apply their mathematical thinking. (810 words)




Sharon McCleary                                                                     3
Mathematics in Middle and Upper Primary                               EDUC8505


References

Baroody, A. & Coslick, R. (1998). Fostering Children’s Mathematical Power, An
      Investigative Approach to K-8 Mathematics Instruction. Lawrence Erlbaum
      Associates, London.

Bottle, G. (2005). Teaching Mathematics in the Primary School. Continuum,
      London.
Burns, M. (2001). Lessons for Introducing Fractions. Math Solution Publications.
      California.

Cathcart, W., Pothier, Y., Vance, J. & Bezuk, N. (2011). Learning Mathematics in
      Elementary and Middle Schools, A Learner-Centered Approach. 5th Edition.
      Pearson Education, Boston.

Clarke, D. & Roche, A. (2009). Students’ fraction comparison strategies as a
      window into robust understanding and possible pointers for instruction.
      DOI: 10.1007/s10649-009-9198-9.
Curriculum Council (Ed.). (1998). Curriculum Framework, Kindergarten to Year 12

      Education in Western Australia (Mathematics Learning
      Area Statement). Curriculum Council of Western Australia. Perth. WA.
      Retrieved from http://www.curriculum.wa.edu.au
Curriculum Council. (2005). Outcomes and Standards Framework and

      Syllabus Documents, Progress Maps and Curriculum Guide. Curriculum
      Council of Western Australia. Perth. WA.
      Retrieved from http://www.curriculum.wa.edu.au
Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference

      Between Epistemological Complementarity and Conflict. Journal for
      Research in Mathematics Education.




Sharon McCleary
             5
Mathematics in Middle and Upper Primary                                   EDUC8505



Copeland, R. (1970). How Children Learn Mathematics, Teaching Implications of
      Piaget’s Research, The Macmillan Company, London.

Cowan, P. (2006). Teaching Mathematics, A Handbook for Primary & Secondary
      School Teachers, Routledge, New York.

Department of Education Victoria. (2011). Fractions and Decimals Online Interview
      Classroom Activities. Retrieved from http://www.education.vic.gov.au
Department of Education and Training Western Australia. (2007). Middle
Childhood:
      Mathematics/ Number Scope and Sequence. Retrieved from:
      http://www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Resources
Devlin, K. (2006). Mathematical Association of America, How do we learn math?.
      Retrieved from: www.maa.org/devlin/devlin_03_06.html

Dienes, Z.P. (1973). Mathematics through the senses, games, dance and art,
      NFER Publishing Company, Ltd, New York.
Downton, A., Knight, R., Clarke, D. & Lewis, G. (2006). Mathematics Assessment

      for Learning: Rich Tasks & Work Samples. Mathematics Teaching and
      Learning Centre. Melbourne. Australia.
Empson, S.B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the co-

      ordination of multiplicatively related quantities: a cross-sectional study of
      children’s thinking. Educational Studies in Mathematics 63, pg1-28.
Flewelling, G., Lind, J. & Sauer, R. (2010). Rich Learning Tasks in Number. The
      Australian Association of Mathematics Teachers. South Australia.

Fraser, C. (2004). The development of the common fraction concept in Grade 3
      learners. Pythagoras 59. pg26-33.

Gray, E., Pitta, D. & Tall, D. (1999). Objects, Actions and Images: A Perspective on
      Early Number Development. Mathematics Education Research Centre,
      Coventry, UK.




Sharon McCleary                                                                       5
Mathematics in Middle and Upper Primary                                   EDUC8505



Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the
      “Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-Year-
      Olds and Adult. Developmental Psychology, Vol. 44, No. 5, pg 1457-1465.

Kamii, C. (1984). Autonomy as the aim of childhood education: A Piagetian
      Approach, Galesburg, IL.

Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic – 2nd Grade-
      Implications of Piaget’s Theory, 2nd Edition, Teachers College Press, London.

Lappan, G., Fey, J., Fitzgerald, W., Friel, S & Phillips, E. (2002). Bits and Pieces I
      Understanding Rational Numbers. Prentice Hall, Illinois.

Lester, F. (Ed.) (2007). Second Handbook of Research on Mathematics Teaching
      and Learning. National Council of Teachers of Mathematics, USA.

McClure, L. (2005). Raising the Profile, Whole School Maths Activities for Primary
      Pupils. The Mathematical Association. Leicester.

McIntosh, A., Reys, B., Reys, R. & Hope, J. (1997). NumberSENSE: Simple Effect
      Number Sense Experiences, Dale Seymour Publications, USA.

Moseley B. (2005). Students’ Early Mathematical Representation Knowledge: The
      Effects of Emphasizing Single or Multiple Perspectives of the Rational
      Number Domain in Problem Solving.
Moss, J. & Case, R. (1999). Developing Children’s Understanding of the Rational

      Numbers: A New Model and an Experimental Curriculum. Journal for
      Research in Mathematics Education. Vol. 30. No. 2. Pp122-47.
Moursand, D., (2006), Mathematics, Retrieved from:
       http://darkwing.uoregon.edu/~moursund/math/mathematics.htm

Muir, T. (2008). Principles of Practice and Teacher Actions: Influences on Effective
      Teaching of Numeracy. Mathematics Education Research Journal. Vol. 20,
      No. 3, pg 78-101.

Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Blackwell Publishers,
      Massachusetts, USA.
Presmeg, N. (n.d.). Research on Visualisation in Learning and Teaching

Sharon McCleary                                                                          6
Mathematics in Middle and Upper Primary                                 EDUC8505

      Mathematics, Illinois State University.
Radford, L., Schubring, G. & Seeger, F. (2011). Signifying and meaning-making in

      mathematical thinking, teaching an learning.
      DOI: 10.1007/s10649-011-9322-5.

Reys, R. & Yang, D.C. (1998). Relationship Between Computational Performance
      and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan.
      Journal for Research in Mathematics Education. Vol. 29, No. 2, pg 225-237.
Schneider, M., Grabner, R. & Paetsch, J. (2009). Mental Number Line, Number

      Line Estimation, and Mathematical Achievement: Their Interrelations in
      Grads 5 and 6. Journal of Educational Psychology. Vol. 101, No. 2. pgs 359-
      372.
Siegler, R., Thompson, C. & Schneider, M. (2011). An Integrated theory of whole

      number and fractions development. Cognitive Psychology. Vol. 62. pp273-
      296.
Smith, C., Solomon, G. & Carey, S. (2005). Never getting to zero: elementary

      school students’ understanding of the infinite divisibility of number and
      matter. Cognitive Psychology. Vol.51. pp101-140.
Stenmark, J. & Bush, W. (2001). Mathematics Assessment, A Practical Handbook.
      National Council of Teachers of Mathematics. VA.

Sullivan, P. & Lilburn, P. (2004). Open-ended Maths Activities, Using ‘good’
      questions to enhance learning in Mathematics. 2nd Edition, Oxford University
      Press, Oxford.
The Australian Curriculum-Mathematics, Version 1.1, (2010). Australian
      Curriculum, Assessment and Reporting Authority [ACARA], Retrieved from:
      http://www.australiancurriculum.edu.au

Vygotsky, L., (1978). Mind in Society, Harvard University Press, Cambridge, MA.
Way, J. & Bobis, J. (2011). Fractions, Teaching for Understanding. The Australian
      Association of Mathematics Teachers Inc. South Australia.

Wilkerson-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?:
      resources and acts for constructing and understanding mathematicians, doi:
      10.1007/s10649-011-9306-5.
Sharon McCleary                                                                      7
Mathematics in Middle and Upper Primary                                 EDUC8505

Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K. (2004),

      First Steps in Mathematics: Number (Book 1). Rigby. Australia.




Sharon McCleary                                                                    8
Mathematics in Middle and Upper Primary                               EDUC8505




References


Baroody, A. & Coslick, R. (1998). Fostering Children’s Mathematical Power, An

      Investigative Approach to K-8 Mathematics Instruction. Lawrence Erlbaum
      Associates, London.
Bottle, G. (2005). Teaching Mathematics in the Primary School. Continuum,
      London.

Burns, M. (2001). Lessons for Introducing Fractions. Math Solution Publications.
      California.
Cathcart, W., Pothier, Y., Vance, J. & Bezuk, N. (2011). Learning Mathematics in

      Elementary and Middle Schools, A Learner-Centered Approach. 5th Edition.
      Pearson Education, Boston.
Clarke, D. & Roche, A. (2009). Students’ fraction comparison strategies as a

      window into robust understanding and possible pointers for instruction.
      DOI: 10.1007/s10649-009-9198-9.

Curriculum Council (Ed.). (1998). Curriculum Framework, Kindergarten to Year 12
      Education in Western Australia (Mathematics Learning
      Area Statement). Curriculum Council of Western Australia. Perth. WA.
      Retrieved from http://www.curriculum.wa.edu.au

Curriculum Council. (2005). Outcomes and Standards Framework and
      Syllabus Documents, Progress Maps and Curriculum Guide. Curriculum
      Council of Western Australia. Perth. WA.
      Retrieved from http://www.curriculum.wa.edu.au

Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference
      Between Epistemological Complementarity and Conflict. Journal for
      Research in Mathematics Education.




Sharon McCleary
             5
Mathematics in Middle and Upper Primary                                   EDUC8505



Copeland, R. (1970). How Children Learn Mathematics, Teaching Implications of
      Piaget’s Research, The Macmillan Company, London.

Cowan, P. (2006). Teaching Mathematics, A Handbook for Primary & Secondary
      School Teachers, Routledge, New York.

Department of Education Victoria. (2011). Fractions and Decimals Online Interview
      Classroom Activities. Retrieved from http://www.education.vic.gov.au
Department of Education and Training Western Australia. (2007). Middle
Childhood:
      Mathematics/ Number Scope and Sequence. Retrieved from:
      http://www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Resources
Devlin, K. (2006). Mathematical Association of America, How do we learn math?.
      Retrieved from: www.maa.org/devlin/devlin_03_06.html

Dienes, Z.P. (1973). Mathematics through the senses, games, dance and art,
      NFER Publishing Company, Ltd, New York.
Downton, A., Knight, R., Clarke, D. & Lewis, G. (2006). Mathematics Assessment

      for Learning: Rich Tasks & Work Samples. Mathematics Teaching and
      Learning Centre. Melbourne. Australia.
Empson, S.B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the co-

      ordination of multiplicatively related quantities: a cross-sectional study of
      children’s thinking. Educational Studies in Mathematics 63, pg1-28.
Flewelling, G., Lind, J. & Sauer, R. (2010). Rich Learning Tasks in Number. The
      Australian Association of Mathematics Teachers. South Australia.

Fraser, C. (2004). The development of the common fraction concept in Grade 3
      learners. Pythagoras 59. pg26-33.

Gray, E., Pitta, D. & Tall, D. (1999). Objects, Actions and Images: A Perspective on
      Early Number Development. Mathematics Education Research Centre,
      Coventry, UK.




Sharon McCleary                                                                       10
Mathematics in Middle and Upper Primary                                   EDUC8505



Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the
      “Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-Year-
      Olds and Adult. Developmental Psychology, Vol. 44, No. 5, pg 1457-1465.

Kamii, C. (1984). Autonomy as the aim of childhood education: A Piagetian
      Approach, Galesburg, IL.

Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic – 2nd Grade-
      Implications of Piaget’s Theory, 2nd Edition, Teachers College Press, London.

Lappan, G., Fey, J., Fitzgerald, W., Friel, S & Phillips, E. (2002). Bits and Pieces I
      Understanding Rational Numbers. Prentice Hall, Illinois.

Lester, F. (Ed.) (2007). Second Handbook of Research on Mathematics Teaching
      and Learning. National Council of Teachers of Mathematics, USA.

McClure, L. (2005). Raising the Profile, Whole School Maths Activities for Primary
      Pupils. The Mathematical Association. Leicester.

McIntosh, A., Reys, B., Reys, R. & Hope, J. (1997). NumberSENSE: Simple Effect
      Number Sense Experiences, Dale Seymour Publications, USA.

Moseley B. (2005). Students’ Early Mathematical Representation Knowledge: The
      Effects of Emphasizing Single or Multiple Perspectives of the Rational
      Number Domain in Problem Solving.
Moss, J. & Case, R. (1999). Developing Children’s Understanding of the Rational

      Numbers: A New Model and an Experimental Curriculum. Journal for
      Research in Mathematics Education. Vol. 30. No. 2. Pp122-47.
Moursand, D., (2006), Mathematics, Retrieved from:
       http://darkwing.uoregon.edu/~moursund/math/mathematics.htm

Muir, T. (2008). Principles of Practice and Teacher Actions: Influences on Effective
      Teaching of Numeracy. Mathematics Education Research Journal. Vol. 20,
      No. 3, pg 78-101.

Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Blackwell Publishers,
      Massachusetts, USA.
Presmeg, N. (n.d.). Research on Visualisation in Learning and Teaching

Sharon McCleary                                                                      11
Mathematics in Middle and Upper Primary                                 EDUC8505

      Mathematics, Illinois State University.
Radford, L., Schubring, G. & Seeger, F. (2011). Signifying and meaning-making in

      mathematical thinking, teaching an learning.
      DOI: 10.1007/s10649-011-9322-5.

Reys, R. & Yang, D.C. (1998). Relationship Between Computational Performance
      and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan.
      Journal for Research in Mathematics Education. Vol. 29, No. 2, pg 225-237.
Schneider, M., Grabner, R. & Paetsch, J. (2009). Mental Number Line, Number

      Line Estimation, and Mathematical Achievement: Their Interrelations in
      Grads 5 and 6. Journal of Educational Psychology. Vol. 101, No. 2. pgs 359-
      372.
Siegler, R., Thompson, C. & Schneider, M. (2011). An Integrated theory of whole

      number and fractions development. Cognitive Psychology. Vol. 62. pp273-
      296.
Smith, C., Solomon, G. & Carey, S. (2005). Never getting to zero: elementary

      school students’ understanding of the infinite divisibility of number and
      matter. Cognitive Psychology. Vol.51. pp101-140.
Stenmark, J. & Bush, W. (2001). Mathematics Assessment, A Practical Handbook.
      National Council of Teachers of Mathematics. VA.

Sullivan, P. & Lilburn, P. (2004). Open-ended Maths Activities, Using ‘good’
      questions to enhance learning in Mathematics. 2nd Edition, Oxford University
      Press, Oxford.
The Australian Curriculum-Mathematics, Version 1.1, (2010). Australian
      Curriculum, Assessment and Reporting Authority [ACARA], Retrieved from:
      http://www.australiancurriculum.edu.au

Vygotsky, L., (1978). Mind in Society, Harvard University Press, Cambridge, MA.
Way, J. & Bobis, J. (2011). Fractions, Teaching for Understanding. The Australian
      Association of Mathematics Teachers Inc. South Australia.

Wilkerson-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?:
      resources and acts for constructing and understanding mathematicians, doi:
      10.1007/s10649-011-9306-5.
Sharon McCleary                                                                    12
Mathematics in Middle and Upper Primary                                 EDUC8505

Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K. (2004),

      First Steps in Mathematics: Number (Book 1). Rigby. Australia.




Sharon McCleary                                                                    13

More Related Content

What's hot

Nature and principles of teaching and learning math
Nature and principles of teaching and learning mathNature and principles of teaching and learning math
Nature and principles of teaching and learning mathJunarie Ramirez
 
Math Curriculum Guide with tagged math equipment
Math Curriculum Guide with tagged math equipmentMath Curriculum Guide with tagged math equipment
Math Curriculum Guide with tagged math equipmentJohndy Ruloma
 
CTET Mathematics Pedagogy part -1
CTET Mathematics Pedagogy part -1CTET Mathematics Pedagogy part -1
CTET Mathematics Pedagogy part -1vaishali chaturvedi
 
Math curriculum guide grades 1 10
Math curriculum guide grades 1 10 Math curriculum guide grades 1 10
Math curriculum guide grades 1 10 Marivic Frias
 
K to-12-mathematic-curriculum-guide-grade-1
K to-12-mathematic-curriculum-guide-grade-1K to-12-mathematic-curriculum-guide-grade-1
K to-12-mathematic-curriculum-guide-grade-1Hannah Karylle Dacillo
 
Math 10 Curriculum Guide rev.2016
Math 10 Curriculum Guide rev.2016Math 10 Curriculum Guide rev.2016
Math 10 Curriculum Guide rev.2016Chuckry Maunes
 
Math 5 Curriculum Guide rev.2016
Math 5 Curriculum Guide rev.2016Math 5 Curriculum Guide rev.2016
Math 5 Curriculum Guide rev.2016Chuckry Maunes
 
Conceptual framework of mathematics
Conceptual framework of mathematicsConceptual framework of mathematics
Conceptual framework of mathematicsAndresBrutas
 
K 12 mathematics standards
K 12 mathematics standardsK 12 mathematics standards
K 12 mathematics standardsjhon basco
 
Ratio and Proportion PD Template
Ratio and Proportion PD TemplateRatio and Proportion PD Template
Ratio and Proportion PD TemplateMatthew Leach
 
K to 12 Mathematics Curriculum Guide for Grades 1 to 10
K to 12 Mathematics Curriculum Guide for Grades 1 to 10K to 12 Mathematics Curriculum Guide for Grades 1 to 10
K to 12 Mathematics Curriculum Guide for Grades 1 to 10Dr. Joy Kenneth Sala Biasong
 
Mathematics Anxiety- Ana
Mathematics Anxiety-  AnaMathematics Anxiety-  Ana
Mathematics Anxiety- AnaAdilah Yahaya
 
Presentatie researchED Amsterdam
Presentatie researchED AmsterdamPresentatie researchED Amsterdam
Presentatie researchED AmsterdamChristian Bokhove
 

What's hot (20)

Philip Siaw Kissi
Philip Siaw KissiPhilip Siaw Kissi
Philip Siaw Kissi
 
Nature and principles of teaching and learning math
Nature and principles of teaching and learning mathNature and principles of teaching and learning math
Nature and principles of teaching and learning math
 
Math Curriculum Guide with tagged math equipment
Math Curriculum Guide with tagged math equipmentMath Curriculum Guide with tagged math equipment
Math Curriculum Guide with tagged math equipment
 
CTET Mathematics Pedagogy part -1
CTET Mathematics Pedagogy part -1CTET Mathematics Pedagogy part -1
CTET Mathematics Pedagogy part -1
 
Lesson guides
Lesson guidesLesson guides
Lesson guides
 
Mathematics K to 12 Curriculum Guide
Mathematics K to 12 Curriculum GuideMathematics K to 12 Curriculum Guide
Mathematics K to 12 Curriculum Guide
 
Math curriculum guide grades 1 10
Math curriculum guide grades 1 10 Math curriculum guide grades 1 10
Math curriculum guide grades 1 10
 
K to-12-mathematic-curriculum-guide-grade-1
K to-12-mathematic-curriculum-guide-grade-1K to-12-mathematic-curriculum-guide-grade-1
K to-12-mathematic-curriculum-guide-grade-1
 
Math 3 kto12 cg
Math 3 kto12  cgMath 3 kto12  cg
Math 3 kto12 cg
 
Math 10 Curriculum Guide rev.2016
Math 10 Curriculum Guide rev.2016Math 10 Curriculum Guide rev.2016
Math 10 Curriculum Guide rev.2016
 
Arithmetic skills
Arithmetic   skillsArithmetic   skills
Arithmetic skills
 
paper #3 for edu 510_fox
paper #3 for edu 510_foxpaper #3 for edu 510_fox
paper #3 for edu 510_fox
 
Math 5 Curriculum Guide rev.2016
Math 5 Curriculum Guide rev.2016Math 5 Curriculum Guide rev.2016
Math 5 Curriculum Guide rev.2016
 
Conceptual framework of mathematics
Conceptual framework of mathematicsConceptual framework of mathematics
Conceptual framework of mathematics
 
K 12 mathematics standards
K 12 mathematics standardsK 12 mathematics standards
K 12 mathematics standards
 
Ratio and Proportion PD Template
Ratio and Proportion PD TemplateRatio and Proportion PD Template
Ratio and Proportion PD Template
 
AMET-NAMA
AMET-NAMAAMET-NAMA
AMET-NAMA
 
K to 12 Mathematics Curriculum Guide for Grades 1 to 10
K to 12 Mathematics Curriculum Guide for Grades 1 to 10K to 12 Mathematics Curriculum Guide for Grades 1 to 10
K to 12 Mathematics Curriculum Guide for Grades 1 to 10
 
Mathematics Anxiety- Ana
Mathematics Anxiety-  AnaMathematics Anxiety-  Ana
Mathematics Anxiety- Ana
 
Presentatie researchED Amsterdam
Presentatie researchED AmsterdamPresentatie researchED Amsterdam
Presentatie researchED Amsterdam
 

Similar to Fractions Rationale

Authentic Tasks And Mathematical Problem Solving
Authentic Tasks And Mathematical Problem SolvingAuthentic Tasks And Mathematical Problem Solving
Authentic Tasks And Mathematical Problem SolvingJim Webb
 
An Analysis Of Challenges In The Teaching Of Problem Solving In Grade 10 Math...
An Analysis Of Challenges In The Teaching Of Problem Solving In Grade 10 Math...An Analysis Of Challenges In The Teaching Of Problem Solving In Grade 10 Math...
An Analysis Of Challenges In The Teaching Of Problem Solving In Grade 10 Math...Courtney Esco
 
TASK 2 DIDACTICS OF MATHEMATICS
TASK 2   DIDACTICS OF MATHEMATICSTASK 2   DIDACTICS OF MATHEMATICS
TASK 2 DIDACTICS OF MATHEMATICSdainerbonilla
 
Fostering Students’ Creativity through Van Hiele’s 5 phase-Based Tangram Acti...
Fostering Students’ Creativity through Van Hiele’s 5 phase-Based Tangram Acti...Fostering Students’ Creativity through Van Hiele’s 5 phase-Based Tangram Acti...
Fostering Students’ Creativity through Van Hiele’s 5 phase-Based Tangram Acti...Chin Lu Chong
 
Secondary level-mathematics-curriculum-of-kerala
Secondary level-mathematics-curriculum-of-keralaSecondary level-mathematics-curriculum-of-kerala
Secondary level-mathematics-curriculum-of-keralakgbiju
 
Understand addition through modelling and manipulation of concrete materials
Understand addition through modelling and manipulation of concrete materialsUnderstand addition through modelling and manipulation of concrete materials
Understand addition through modelling and manipulation of concrete materialsAlexander Decker
 
Algebraic Thinking A Problem Solving Approach
Algebraic Thinking  A Problem Solving ApproachAlgebraic Thinking  A Problem Solving Approach
Algebraic Thinking A Problem Solving ApproachCheryl Brown
 
Building research and development partnerships between schools and Higher Edu...
Building research and development partnerships between schools and Higher Edu...Building research and development partnerships between schools and Higher Edu...
Building research and development partnerships between schools and Higher Edu...Brian Hudson
 
Assessing Multiplicative Thinking Using Rich Tasks
Assessing Multiplicative Thinking Using Rich TasksAssessing Multiplicative Thinking Using Rich Tasks
Assessing Multiplicative Thinking Using Rich TasksMaria Perkins
 
A test of the efficacy of field trip and discussion approaches to teaching in...
A test of the efficacy of field trip and discussion approaches to teaching in...A test of the efficacy of field trip and discussion approaches to teaching in...
A test of the efficacy of field trip and discussion approaches to teaching in...Alexander Decker
 
Exploring 8th Grade Middle School Science Teachers’ Use of Web 2.0 Tools
Exploring 8th Grade Middle School Science Teachers’ Use of Web 2.0 ToolsExploring 8th Grade Middle School Science Teachers’ Use of Web 2.0 Tools
Exploring 8th Grade Middle School Science Teachers’ Use of Web 2.0 ToolsAntwuan Stinson
 
Edx3270 literacies education assignment one
Edx3270 literacies education assignment oneEdx3270 literacies education assignment one
Edx3270 literacies education assignment onemdskc5966
 
Number Recognition Assignment
Number Recognition AssignmentNumber Recognition Assignment
Number Recognition AssignmentPaula Smith
 
A General Analytical Model For Problem Solving Teaching BoS
A General Analytical Model For Problem Solving Teaching  BoSA General Analytical Model For Problem Solving Teaching  BoS
A General Analytical Model For Problem Solving Teaching BoSDereck Downing
 
Enhancing Pupils’ Knowledge of Mathematical Concepts through Game and Poem
Enhancing Pupils’ Knowledge of Mathematical Concepts through Game and PoemEnhancing Pupils’ Knowledge of Mathematical Concepts through Game and Poem
Enhancing Pupils’ Knowledge of Mathematical Concepts through Game and Poemiosrjce
 
MATD611 Mathematics Education In Perspective.docx
MATD611 Mathematics Education In Perspective.docxMATD611 Mathematics Education In Perspective.docx
MATD611 Mathematics Education In Perspective.docxstirlingvwriters
 

Similar to Fractions Rationale (20)

Authentic Tasks And Mathematical Problem Solving
Authentic Tasks And Mathematical Problem SolvingAuthentic Tasks And Mathematical Problem Solving
Authentic Tasks And Mathematical Problem Solving
 
An Analysis Of Challenges In The Teaching Of Problem Solving In Grade 10 Math...
An Analysis Of Challenges In The Teaching Of Problem Solving In Grade 10 Math...An Analysis Of Challenges In The Teaching Of Problem Solving In Grade 10 Math...
An Analysis Of Challenges In The Teaching Of Problem Solving In Grade 10 Math...
 
TASK 2 DIDACTICS OF MATHEMATICS
TASK 2   DIDACTICS OF MATHEMATICSTASK 2   DIDACTICS OF MATHEMATICS
TASK 2 DIDACTICS OF MATHEMATICS
 
Paper Folding
Paper FoldingPaper Folding
Paper Folding
 
Jan 2013
Jan 2013Jan 2013
Jan 2013
 
Fostering Students’ Creativity through Van Hiele’s 5 phase-Based Tangram Acti...
Fostering Students’ Creativity through Van Hiele’s 5 phase-Based Tangram Acti...Fostering Students’ Creativity through Van Hiele’s 5 phase-Based Tangram Acti...
Fostering Students’ Creativity through Van Hiele’s 5 phase-Based Tangram Acti...
 
Secondary level-mathematics-curriculum-of-kerala
Secondary level-mathematics-curriculum-of-keralaSecondary level-mathematics-curriculum-of-kerala
Secondary level-mathematics-curriculum-of-kerala
 
Understand addition through modelling and manipulation of concrete materials
Understand addition through modelling and manipulation of concrete materialsUnderstand addition through modelling and manipulation of concrete materials
Understand addition through modelling and manipulation of concrete materials
 
Mathematics CG 2023.pdf
Mathematics CG 2023.pdfMathematics CG 2023.pdf
Mathematics CG 2023.pdf
 
Algebraic Thinking A Problem Solving Approach
Algebraic Thinking  A Problem Solving ApproachAlgebraic Thinking  A Problem Solving Approach
Algebraic Thinking A Problem Solving Approach
 
Building research and development partnerships between schools and Higher Edu...
Building research and development partnerships between schools and Higher Edu...Building research and development partnerships between schools and Higher Edu...
Building research and development partnerships between schools and Higher Edu...
 
Assessing Multiplicative Thinking Using Rich Tasks
Assessing Multiplicative Thinking Using Rich TasksAssessing Multiplicative Thinking Using Rich Tasks
Assessing Multiplicative Thinking Using Rich Tasks
 
A test of the efficacy of field trip and discussion approaches to teaching in...
A test of the efficacy of field trip and discussion approaches to teaching in...A test of the efficacy of field trip and discussion approaches to teaching in...
A test of the efficacy of field trip and discussion approaches to teaching in...
 
Exploring 8th Grade Middle School Science Teachers’ Use of Web 2.0 Tools
Exploring 8th Grade Middle School Science Teachers’ Use of Web 2.0 ToolsExploring 8th Grade Middle School Science Teachers’ Use of Web 2.0 Tools
Exploring 8th Grade Middle School Science Teachers’ Use of Web 2.0 Tools
 
Edx3270 literacies education assignment one
Edx3270 literacies education assignment oneEdx3270 literacies education assignment one
Edx3270 literacies education assignment one
 
Number Recognition Assignment
Number Recognition AssignmentNumber Recognition Assignment
Number Recognition Assignment
 
A General Analytical Model For Problem Solving Teaching BoS
A General Analytical Model For Problem Solving Teaching  BoSA General Analytical Model For Problem Solving Teaching  BoS
A General Analytical Model For Problem Solving Teaching BoS
 
AAMT Connect with Maths webinar: The importance of talk for mathematical lear...
AAMT Connect with Maths webinar: The importance of talk for mathematical lear...AAMT Connect with Maths webinar: The importance of talk for mathematical lear...
AAMT Connect with Maths webinar: The importance of talk for mathematical lear...
 
Enhancing Pupils’ Knowledge of Mathematical Concepts through Game and Poem
Enhancing Pupils’ Knowledge of Mathematical Concepts through Game and PoemEnhancing Pupils’ Knowledge of Mathematical Concepts through Game and Poem
Enhancing Pupils’ Knowledge of Mathematical Concepts through Game and Poem
 
MATD611 Mathematics Education In Perspective.docx
MATD611 Mathematics Education In Perspective.docxMATD611 Mathematics Education In Perspective.docx
MATD611 Mathematics Education In Perspective.docx
 

More from sharon-mccleary

Think Board Interview, Recommendations and Reflection
Think Board Interview, Recommendations and ReflectionThink Board Interview, Recommendations and Reflection
Think Board Interview, Recommendations and Reflectionsharon-mccleary
 
Think Board Interview, Recommendations and Reflection
Think Board Interview, Recommendations and ReflectionThink Board Interview, Recommendations and Reflection
Think Board Interview, Recommendations and Reflectionsharon-mccleary
 
Analysis of Spelling Data - A Case Study
Analysis of Spelling Data - A Case StudyAnalysis of Spelling Data - A Case Study
Analysis of Spelling Data - A Case Studysharon-mccleary
 
Stay in Step Follow Up Lesson Plan
Stay in Step Follow Up Lesson PlanStay in Step Follow Up Lesson Plan
Stay in Step Follow Up Lesson Plansharon-mccleary
 
Stay in Step FMS Assessment Report
Stay in Step FMS Assessment ReportStay in Step FMS Assessment Report
Stay in Step FMS Assessment Reportsharon-mccleary
 
Fractions Activity Outlines
Fractions Activity OutlinesFractions Activity Outlines
Fractions Activity Outlinessharon-mccleary
 
SOSE Programme: Recognition, Respect, Reconciliation
SOSE Programme: Recognition, Respect, ReconciliationSOSE Programme: Recognition, Respect, Reconciliation
SOSE Programme: Recognition, Respect, Reconciliationsharon-mccleary
 

More from sharon-mccleary (16)

Think Board Interview, Recommendations and Reflection
Think Board Interview, Recommendations and ReflectionThink Board Interview, Recommendations and Reflection
Think Board Interview, Recommendations and Reflection
 
Think Board Interview, Recommendations and Reflection
Think Board Interview, Recommendations and ReflectionThink Board Interview, Recommendations and Reflection
Think Board Interview, Recommendations and Reflection
 
Sample Spelling Plans
Sample Spelling PlansSample Spelling Plans
Sample Spelling Plans
 
Analysis of Spelling Data - A Case Study
Analysis of Spelling Data - A Case StudyAnalysis of Spelling Data - A Case Study
Analysis of Spelling Data - A Case Study
 
Stay in Step Follow Up Lesson Plan
Stay in Step Follow Up Lesson PlanStay in Step Follow Up Lesson Plan
Stay in Step Follow Up Lesson Plan
 
Stay in Step FMS Assessment Report
Stay in Step FMS Assessment ReportStay in Step FMS Assessment Report
Stay in Step FMS Assessment Report
 
Kidsafe booklet
Kidsafe bookletKidsafe booklet
Kidsafe booklet
 
Fractions Activity Outlines
Fractions Activity OutlinesFractions Activity Outlines
Fractions Activity Outlines
 
Pilbara Resource File
Pilbara Resource FilePilbara Resource File
Pilbara Resource File
 
SOSE Programme: Recognition, Respect, Reconciliation
SOSE Programme: Recognition, Respect, ReconciliationSOSE Programme: Recognition, Respect, Reconciliation
SOSE Programme: Recognition, Respect, Reconciliation
 
Main idea 1
Main idea 1Main idea 1
Main idea 1
 
Main idea 2
Main idea 2Main idea 2
Main idea 2
 
Main idea 3
Main idea 3Main idea 3
Main idea 3
 
Main idea 4
Main idea 4Main idea 4
Main idea 4
 
Fractions programme
Fractions programmeFractions programme
Fractions programme
 
Energy Lesson Plans
Energy Lesson PlansEnergy Lesson Plans
Energy Lesson Plans
 

Recently uploaded

ARTERIAL BLOOD GAS ANALYSIS........pptx
ARTERIAL BLOOD  GAS ANALYSIS........pptxARTERIAL BLOOD  GAS ANALYSIS........pptx
ARTERIAL BLOOD GAS ANALYSIS........pptxAneriPatwari
 
ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfVanessa Camilleri
 
Congestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationCongestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationdeepaannamalai16
 
Unraveling Hypertext_ Analyzing Postmodern Elements in Literature.pptx
Unraveling Hypertext_ Analyzing  Postmodern Elements in  Literature.pptxUnraveling Hypertext_ Analyzing  Postmodern Elements in  Literature.pptx
Unraveling Hypertext_ Analyzing Postmodern Elements in Literature.pptxDhatriParmar
 
4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptxmary850239
 
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...DhatriParmar
 
Q-Factor General Quiz-7th April 2024, Quiz Club NITW
Q-Factor General Quiz-7th April 2024, Quiz Club NITWQ-Factor General Quiz-7th April 2024, Quiz Club NITW
Q-Factor General Quiz-7th April 2024, Quiz Club NITWQuiz Club NITW
 
Indexing Structures in Database Management system.pdf
Indexing Structures in Database Management system.pdfIndexing Structures in Database Management system.pdf
Indexing Structures in Database Management system.pdfChristalin Nelson
 
31 ĐỀ THI THỬ VÀO LỚP 10 - TIẾNG ANH - FORM MỚI 2025 - 40 CÂU HỎI - BÙI VĂN V...
31 ĐỀ THI THỬ VÀO LỚP 10 - TIẾNG ANH - FORM MỚI 2025 - 40 CÂU HỎI - BÙI VĂN V...31 ĐỀ THI THỬ VÀO LỚP 10 - TIẾNG ANH - FORM MỚI 2025 - 40 CÂU HỎI - BÙI VĂN V...
31 ĐỀ THI THỬ VÀO LỚP 10 - TIẾNG ANH - FORM MỚI 2025 - 40 CÂU HỎI - BÙI VĂN V...Nguyen Thanh Tu Collection
 
Oppenheimer Film Discussion for Philosophy and Film
Oppenheimer Film Discussion for Philosophy and FilmOppenheimer Film Discussion for Philosophy and Film
Oppenheimer Film Discussion for Philosophy and FilmStan Meyer
 
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptxDIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptxMichelleTuguinay1
 
4.11.24 Mass Incarceration and the New Jim Crow.pptx
4.11.24 Mass Incarceration and the New Jim Crow.pptx4.11.24 Mass Incarceration and the New Jim Crow.pptx
4.11.24 Mass Incarceration and the New Jim Crow.pptxmary850239
 
CLASSIFICATION OF ANTI - CANCER DRUGS.pptx
CLASSIFICATION OF ANTI - CANCER DRUGS.pptxCLASSIFICATION OF ANTI - CANCER DRUGS.pptx
CLASSIFICATION OF ANTI - CANCER DRUGS.pptxAnupam32727
 
Expanded definition: technical and operational
Expanded definition: technical and operationalExpanded definition: technical and operational
Expanded definition: technical and operationalssuser3e220a
 
Reading and Writing Skills 11 quarter 4 melc 1
Reading and Writing Skills 11 quarter 4 melc 1Reading and Writing Skills 11 quarter 4 melc 1
Reading and Writing Skills 11 quarter 4 melc 1GloryAnnCastre1
 
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITWQ-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITWQuiz Club NITW
 
Daily Lesson Plan in Mathematics Quarter 4
Daily Lesson Plan in Mathematics Quarter 4Daily Lesson Plan in Mathematics Quarter 4
Daily Lesson Plan in Mathematics Quarter 4JOYLYNSAMANIEGO
 
How to Make a Duplicate of Your Odoo 17 Database
How to Make a Duplicate of Your Odoo 17 DatabaseHow to Make a Duplicate of Your Odoo 17 Database
How to Make a Duplicate of Your Odoo 17 DatabaseCeline George
 
Decoding the Tweet _ Practical Criticism in the Age of Hashtag.pptx
Decoding the Tweet _ Practical Criticism in the Age of Hashtag.pptxDecoding the Tweet _ Practical Criticism in the Age of Hashtag.pptx
Decoding the Tweet _ Practical Criticism in the Age of Hashtag.pptxDhatriParmar
 

Recently uploaded (20)

ARTERIAL BLOOD GAS ANALYSIS........pptx
ARTERIAL BLOOD  GAS ANALYSIS........pptxARTERIAL BLOOD  GAS ANALYSIS........pptx
ARTERIAL BLOOD GAS ANALYSIS........pptx
 
ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdf
 
Congestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationCongestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentation
 
Unraveling Hypertext_ Analyzing Postmodern Elements in Literature.pptx
Unraveling Hypertext_ Analyzing  Postmodern Elements in  Literature.pptxUnraveling Hypertext_ Analyzing  Postmodern Elements in  Literature.pptx
Unraveling Hypertext_ Analyzing Postmodern Elements in Literature.pptx
 
4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx
 
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
Blowin' in the Wind of Caste_ Bob Dylan's Song as a Catalyst for Social Justi...
 
Q-Factor General Quiz-7th April 2024, Quiz Club NITW
Q-Factor General Quiz-7th April 2024, Quiz Club NITWQ-Factor General Quiz-7th April 2024, Quiz Club NITW
Q-Factor General Quiz-7th April 2024, Quiz Club NITW
 
Indexing Structures in Database Management system.pdf
Indexing Structures in Database Management system.pdfIndexing Structures in Database Management system.pdf
Indexing Structures in Database Management system.pdf
 
31 ĐỀ THI THỬ VÀO LỚP 10 - TIẾNG ANH - FORM MỚI 2025 - 40 CÂU HỎI - BÙI VĂN V...
31 ĐỀ THI THỬ VÀO LỚP 10 - TIẾNG ANH - FORM MỚI 2025 - 40 CÂU HỎI - BÙI VĂN V...31 ĐỀ THI THỬ VÀO LỚP 10 - TIẾNG ANH - FORM MỚI 2025 - 40 CÂU HỎI - BÙI VĂN V...
31 ĐỀ THI THỬ VÀO LỚP 10 - TIẾNG ANH - FORM MỚI 2025 - 40 CÂU HỎI - BÙI VĂN V...
 
Oppenheimer Film Discussion for Philosophy and Film
Oppenheimer Film Discussion for Philosophy and FilmOppenheimer Film Discussion for Philosophy and Film
Oppenheimer Film Discussion for Philosophy and Film
 
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptxDIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
DIFFERENT BASKETRY IN THE PHILIPPINES PPT.pptx
 
4.11.24 Mass Incarceration and the New Jim Crow.pptx
4.11.24 Mass Incarceration and the New Jim Crow.pptx4.11.24 Mass Incarceration and the New Jim Crow.pptx
4.11.24 Mass Incarceration and the New Jim Crow.pptx
 
CLASSIFICATION OF ANTI - CANCER DRUGS.pptx
CLASSIFICATION OF ANTI - CANCER DRUGS.pptxCLASSIFICATION OF ANTI - CANCER DRUGS.pptx
CLASSIFICATION OF ANTI - CANCER DRUGS.pptx
 
Expanded definition: technical and operational
Expanded definition: technical and operationalExpanded definition: technical and operational
Expanded definition: technical and operational
 
Reading and Writing Skills 11 quarter 4 melc 1
Reading and Writing Skills 11 quarter 4 melc 1Reading and Writing Skills 11 quarter 4 melc 1
Reading and Writing Skills 11 quarter 4 melc 1
 
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITWQ-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
Q-Factor HISPOL Quiz-6th April 2024, Quiz Club NITW
 
Daily Lesson Plan in Mathematics Quarter 4
Daily Lesson Plan in Mathematics Quarter 4Daily Lesson Plan in Mathematics Quarter 4
Daily Lesson Plan in Mathematics Quarter 4
 
How to Make a Duplicate of Your Odoo 17 Database
How to Make a Duplicate of Your Odoo 17 DatabaseHow to Make a Duplicate of Your Odoo 17 Database
How to Make a Duplicate of Your Odoo 17 Database
 
Decoding the Tweet _ Practical Criticism in the Age of Hashtag.pptx
Decoding the Tweet _ Practical Criticism in the Age of Hashtag.pptxDecoding the Tweet _ Practical Criticism in the Age of Hashtag.pptx
Decoding the Tweet _ Practical Criticism in the Age of Hashtag.pptx
 
prashanth updated resume 2024 for Teaching Profession
prashanth updated resume 2024 for Teaching Professionprashanth updated resume 2024 for Teaching Profession
prashanth updated resume 2024 for Teaching Profession
 

Fractions Rationale

  • 1. Mathematics in Middle and Upper Primary Fall EDUC8505 08 Rationale Mathematics is a discipline which has evolved from the human need to measure and communicate about time, quantity and space (Moursund, 2002). It is inherently abstract, applicable over a wide field and uses symbols to represent mathematical concepts. Traditional theoretical frameworks associated with children’s mathematical thinking include empiricism, where knowledge is external and acquired through the senses, (neo)nativism/rationalism which emphasises the in-born capabilities of the child to reason, and interactionalism, which recognises interacting roles of nature and experience, and considers the child as active in knowledge construction (Lester, 2007). A central part of each of these frameworks is experiences, which allow children to internalise or express knowledge. Experiences provide opportunities to learn, which are considered “the single most important predictor of student achievement” (National Research Council, 2001, p334; cited in Lester, 2007), and allow children to acquire physical, socio-conventional and logico-mathematical knowledge (Piaget, 1967, cited in Kamii, 2004). They are instrumental in supporting student affect, which plays a crucial role in mathematics teaching and learning (Hart&Walker, 1993, cited in Baroody, 1998). Experiences should illustrate a wide variety of examples relating to the key concept in different contexts, to facilitate students forming multiple representations and connections, and building conceptual understanding, rather than simply applying procedural knowledge. Brownell (cited in Lester, 2007) refers to conceptual understanding as mental connections among mathematical facts, procedures and ideas. Vergnaud (1983, cited in Lester, 2007) introduced the concept of the multiplicative conceptual field, a complex system of interrelated concepts, student ideas (competencies and misconceptions), procedures, problems, representations, objects, properties and relationships that cannot be studied in isolation, including multiplication, division, fractions, ratios, simple and multiple proportions, rational numbers, dimensional analysis and vector spaces. The programme provides varied activities representing the five sub-constructs of fractions (Kieren, 1980, cited in Way Sharon McCleary 5
  • 2. Mathematics in Middle and Upper Primary EDUC8505 & Bobis, 2011) part-whole, measure, quotient, operator, ratio), and encourages connections between different rational numbers (e.g. Lesson 6: Fractions as quotients, incorporating relational concepts and using numbers with common factors, which support richer interconnections (Empson, 2005)). Planned experiences should also aim to expose misconceptions, prevent the formation of new ones (Bottle, 2005), and cater to students of different ability levels by using open-ended activities. The programme achieves this by selecting activities that target common misconceptions and require students to disprove them with concrete materials (e.g. Lesson 3: Show Me A Half). Concrete materials are central to assisting students in the concrete operations phase of development understand mathematical concepts (Kamii, 2004). However their use does not automatically result in mathematical learning, as students can focus on unintended aspects and fail to abstract the intended concept (Gray, 1999). Consequently, opportunities for exploring the manipulative before using it are given (Lesson 4: Exploring Pattern Blocks), and use is closely aligned with conceptual understanding and linked to symbolic conventions in order to promote purposeful connections in students’ minds. Several frameworks characterise mathematical learning as progressing from physical/concrete interaction, to generalising abstract ideas/concepts and representing them symbolically (Cowan, 2006; Lester, 2007; Baroody, 1998). Visual imagery constructed from concrete experiences is central to this progression, and its role in assisting learning has been addressed by several researchers, including Presmeg, Goldin and Thomas. Consequently, several experiences in the programme use concrete materials to encourage clear visual images that may assist children in thinking mathematically (e.g. Lesson 1: water in glasses, Lessons 4&5: Pattern Blocks). Correct mathematical language and writing conventions are also crucial to this process, since effective communication is pivotal in clarifying inconsistencies between the child’s inner understandings and correct conceptual understanding, and in allowing opportunities for exchanges between peers, expanding strategy knowledge through social learning (Vygotsky, 1978). The teacher’s role is to provide clear links Sharon McCleary 2
  • 3. Mathematics in Middle and Upper Primary EDUC8505 between concepts and conventional language/symbols, enabling semiotic meaning making without stifling inherent thought processes. Opportunities to build fluency are provided in each lesson of the programme through reading, writing, talking and listening. Another feature linked to developing conceptual understanding is allowing students to actively expend effort in making sense of important mathematical ideas. Festinger’s (1957) theory of cognitive dissonance describes perplexity as a central impetus for cognitive growth, and Hatano (1988) identifies cognitive incongruity as the critical trigger for developing reasoning skills that display conceptual understanding (Lester, 2007). This is consistent with constructivist ideas of presenting problems near the boundary of the student’s Zone of Proximal Development (Vygotsky, 1978), allowing sufficient challenge to promote thinking and application of conceptual knowledge, while supporting opportunities for success and maintaining positive affect: “Acquired knowledge is most useful to a learner when it is discovered through their own cognitive efforts, related to and used in reference to what one has known before” (Bruner, cited in Cowan, 2006, pg 26). The programme incorporates problem solving allowing different solution methods: Lessons 6 & 12 provide additive and multiplicative thinking arising from invented strategies for division questions, allowing students opportunities to ‘struggle’ with relevant mathematical concepts in authentic scenarios. The teacher’s role encompasses providing children with engaging, challenging and enjoyable experiences which emphasise conceptual understanding and promote a positive attitude towards mathematics. Implicit in this is creating a classroom environment which allows opportunities for discussion, assists students in becoming fluent with conventional mathematical language/symbols and is accepting of invented strategies and solution methods. This facilitates students’ forming connections between multiple representations and abstracting meaning from experiences to progress and apply their mathematical thinking. (810 words) Sharon McCleary 3
  • 4. Mathematics in Middle and Upper Primary EDUC8505 References Baroody, A. & Coslick, R. (1998). Fostering Children’s Mathematical Power, An Investigative Approach to K-8 Mathematics Instruction. Lawrence Erlbaum Associates, London. Bottle, G. (2005). Teaching Mathematics in the Primary School. Continuum, London. Burns, M. (2001). Lessons for Introducing Fractions. Math Solution Publications. California. Cathcart, W., Pothier, Y., Vance, J. & Bezuk, N. (2011). Learning Mathematics in Elementary and Middle Schools, A Learner-Centered Approach. 5th Edition. Pearson Education, Boston. Clarke, D. & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. DOI: 10.1007/s10649-009-9198-9. Curriculum Council (Ed.). (1998). Curriculum Framework, Kindergarten to Year 12 Education in Western Australia (Mathematics Learning Area Statement). Curriculum Council of Western Australia. Perth. WA. Retrieved from http://www.curriculum.wa.edu.au Curriculum Council. (2005). Outcomes and Standards Framework and Syllabus Documents, Progress Maps and Curriculum Guide. Curriculum Council of Western Australia. Perth. WA. Retrieved from http://www.curriculum.wa.edu.au Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference Between Epistemological Complementarity and Conflict. Journal for Research in Mathematics Education. Sharon McCleary 5
  • 5. Mathematics in Middle and Upper Primary EDUC8505 Copeland, R. (1970). How Children Learn Mathematics, Teaching Implications of Piaget’s Research, The Macmillan Company, London. Cowan, P. (2006). Teaching Mathematics, A Handbook for Primary & Secondary School Teachers, Routledge, New York. Department of Education Victoria. (2011). Fractions and Decimals Online Interview Classroom Activities. Retrieved from http://www.education.vic.gov.au Department of Education and Training Western Australia. (2007). Middle Childhood: Mathematics/ Number Scope and Sequence. Retrieved from: http://www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Resources Devlin, K. (2006). Mathematical Association of America, How do we learn math?. Retrieved from: www.maa.org/devlin/devlin_03_06.html Dienes, Z.P. (1973). Mathematics through the senses, games, dance and art, NFER Publishing Company, Ltd, New York. Downton, A., Knight, R., Clarke, D. & Lewis, G. (2006). Mathematics Assessment for Learning: Rich Tasks & Work Samples. Mathematics Teaching and Learning Centre. Melbourne. Australia. Empson, S.B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the co- ordination of multiplicatively related quantities: a cross-sectional study of children’s thinking. Educational Studies in Mathematics 63, pg1-28. Flewelling, G., Lind, J. & Sauer, R. (2010). Rich Learning Tasks in Number. The Australian Association of Mathematics Teachers. South Australia. Fraser, C. (2004). The development of the common fraction concept in Grade 3 learners. Pythagoras 59. pg26-33. Gray, E., Pitta, D. & Tall, D. (1999). Objects, Actions and Images: A Perspective on Early Number Development. Mathematics Education Research Centre, Coventry, UK. Sharon McCleary 5
  • 6. Mathematics in Middle and Upper Primary EDUC8505 Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the “Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-Year- Olds and Adult. Developmental Psychology, Vol. 44, No. 5, pg 1457-1465. Kamii, C. (1984). Autonomy as the aim of childhood education: A Piagetian Approach, Galesburg, IL. Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic – 2nd Grade- Implications of Piaget’s Theory, 2nd Edition, Teachers College Press, London. Lappan, G., Fey, J., Fitzgerald, W., Friel, S & Phillips, E. (2002). Bits and Pieces I Understanding Rational Numbers. Prentice Hall, Illinois. Lester, F. (Ed.) (2007). Second Handbook of Research on Mathematics Teaching and Learning. National Council of Teachers of Mathematics, USA. McClure, L. (2005). Raising the Profile, Whole School Maths Activities for Primary Pupils. The Mathematical Association. Leicester. McIntosh, A., Reys, B., Reys, R. & Hope, J. (1997). NumberSENSE: Simple Effect Number Sense Experiences, Dale Seymour Publications, USA. Moseley B. (2005). Students’ Early Mathematical Representation Knowledge: The Effects of Emphasizing Single or Multiple Perspectives of the Rational Number Domain in Problem Solving. Moss, J. & Case, R. (1999). Developing Children’s Understanding of the Rational Numbers: A New Model and an Experimental Curriculum. Journal for Research in Mathematics Education. Vol. 30. No. 2. Pp122-47. Moursand, D., (2006), Mathematics, Retrieved from: http://darkwing.uoregon.edu/~moursund/math/mathematics.htm Muir, T. (2008). Principles of Practice and Teacher Actions: Influences on Effective Teaching of Numeracy. Mathematics Education Research Journal. Vol. 20, No. 3, pg 78-101. Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Blackwell Publishers, Massachusetts, USA. Presmeg, N. (n.d.). Research on Visualisation in Learning and Teaching Sharon McCleary 6
  • 7. Mathematics in Middle and Upper Primary EDUC8505 Mathematics, Illinois State University. Radford, L., Schubring, G. & Seeger, F. (2011). Signifying and meaning-making in mathematical thinking, teaching an learning. DOI: 10.1007/s10649-011-9322-5. Reys, R. & Yang, D.C. (1998). Relationship Between Computational Performance and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan. Journal for Research in Mathematics Education. Vol. 29, No. 2, pg 225-237. Schneider, M., Grabner, R. & Paetsch, J. (2009). Mental Number Line, Number Line Estimation, and Mathematical Achievement: Their Interrelations in Grads 5 and 6. Journal of Educational Psychology. Vol. 101, No. 2. pgs 359- 372. Siegler, R., Thompson, C. & Schneider, M. (2011). An Integrated theory of whole number and fractions development. Cognitive Psychology. Vol. 62. pp273- 296. Smith, C., Solomon, G. & Carey, S. (2005). Never getting to zero: elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology. Vol.51. pp101-140. Stenmark, J. & Bush, W. (2001). Mathematics Assessment, A Practical Handbook. National Council of Teachers of Mathematics. VA. Sullivan, P. & Lilburn, P. (2004). Open-ended Maths Activities, Using ‘good’ questions to enhance learning in Mathematics. 2nd Edition, Oxford University Press, Oxford. The Australian Curriculum-Mathematics, Version 1.1, (2010). Australian Curriculum, Assessment and Reporting Authority [ACARA], Retrieved from: http://www.australiancurriculum.edu.au Vygotsky, L., (1978). Mind in Society, Harvard University Press, Cambridge, MA. Way, J. & Bobis, J. (2011). Fractions, Teaching for Understanding. The Australian Association of Mathematics Teachers Inc. South Australia. Wilkerson-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?: resources and acts for constructing and understanding mathematicians, doi: 10.1007/s10649-011-9306-5. Sharon McCleary 7
  • 8. Mathematics in Middle and Upper Primary EDUC8505 Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K. (2004), First Steps in Mathematics: Number (Book 1). Rigby. Australia. Sharon McCleary 8
  • 9. Mathematics in Middle and Upper Primary EDUC8505 References Baroody, A. & Coslick, R. (1998). Fostering Children’s Mathematical Power, An Investigative Approach to K-8 Mathematics Instruction. Lawrence Erlbaum Associates, London. Bottle, G. (2005). Teaching Mathematics in the Primary School. Continuum, London. Burns, M. (2001). Lessons for Introducing Fractions. Math Solution Publications. California. Cathcart, W., Pothier, Y., Vance, J. & Bezuk, N. (2011). Learning Mathematics in Elementary and Middle Schools, A Learner-Centered Approach. 5th Edition. Pearson Education, Boston. Clarke, D. & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. DOI: 10.1007/s10649-009-9198-9. Curriculum Council (Ed.). (1998). Curriculum Framework, Kindergarten to Year 12 Education in Western Australia (Mathematics Learning Area Statement). Curriculum Council of Western Australia. Perth. WA. Retrieved from http://www.curriculum.wa.edu.au Curriculum Council. (2005). Outcomes and Standards Framework and Syllabus Documents, Progress Maps and Curriculum Guide. Curriculum Council of Western Australia. Perth. WA. Retrieved from http://www.curriculum.wa.edu.au Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference Between Epistemological Complementarity and Conflict. Journal for Research in Mathematics Education. Sharon McCleary 5
  • 10. Mathematics in Middle and Upper Primary EDUC8505 Copeland, R. (1970). How Children Learn Mathematics, Teaching Implications of Piaget’s Research, The Macmillan Company, London. Cowan, P. (2006). Teaching Mathematics, A Handbook for Primary & Secondary School Teachers, Routledge, New York. Department of Education Victoria. (2011). Fractions and Decimals Online Interview Classroom Activities. Retrieved from http://www.education.vic.gov.au Department of Education and Training Western Australia. (2007). Middle Childhood: Mathematics/ Number Scope and Sequence. Retrieved from: http://www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Resources Devlin, K. (2006). Mathematical Association of America, How do we learn math?. Retrieved from: www.maa.org/devlin/devlin_03_06.html Dienes, Z.P. (1973). Mathematics through the senses, games, dance and art, NFER Publishing Company, Ltd, New York. Downton, A., Knight, R., Clarke, D. & Lewis, G. (2006). Mathematics Assessment for Learning: Rich Tasks & Work Samples. Mathematics Teaching and Learning Centre. Melbourne. Australia. Empson, S.B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the co- ordination of multiplicatively related quantities: a cross-sectional study of children’s thinking. Educational Studies in Mathematics 63, pg1-28. Flewelling, G., Lind, J. & Sauer, R. (2010). Rich Learning Tasks in Number. The Australian Association of Mathematics Teachers. South Australia. Fraser, C. (2004). The development of the common fraction concept in Grade 3 learners. Pythagoras 59. pg26-33. Gray, E., Pitta, D. & Tall, D. (1999). Objects, Actions and Images: A Perspective on Early Number Development. Mathematics Education Research Centre, Coventry, UK. Sharon McCleary 10
  • 11. Mathematics in Middle and Upper Primary EDUC8505 Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the “Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-Year- Olds and Adult. Developmental Psychology, Vol. 44, No. 5, pg 1457-1465. Kamii, C. (1984). Autonomy as the aim of childhood education: A Piagetian Approach, Galesburg, IL. Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic – 2nd Grade- Implications of Piaget’s Theory, 2nd Edition, Teachers College Press, London. Lappan, G., Fey, J., Fitzgerald, W., Friel, S & Phillips, E. (2002). Bits and Pieces I Understanding Rational Numbers. Prentice Hall, Illinois. Lester, F. (Ed.) (2007). Second Handbook of Research on Mathematics Teaching and Learning. National Council of Teachers of Mathematics, USA. McClure, L. (2005). Raising the Profile, Whole School Maths Activities for Primary Pupils. The Mathematical Association. Leicester. McIntosh, A., Reys, B., Reys, R. & Hope, J. (1997). NumberSENSE: Simple Effect Number Sense Experiences, Dale Seymour Publications, USA. Moseley B. (2005). Students’ Early Mathematical Representation Knowledge: The Effects of Emphasizing Single or Multiple Perspectives of the Rational Number Domain in Problem Solving. Moss, J. & Case, R. (1999). Developing Children’s Understanding of the Rational Numbers: A New Model and an Experimental Curriculum. Journal for Research in Mathematics Education. Vol. 30. No. 2. Pp122-47. Moursand, D., (2006), Mathematics, Retrieved from: http://darkwing.uoregon.edu/~moursund/math/mathematics.htm Muir, T. (2008). Principles of Practice and Teacher Actions: Influences on Effective Teaching of Numeracy. Mathematics Education Research Journal. Vol. 20, No. 3, pg 78-101. Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Blackwell Publishers, Massachusetts, USA. Presmeg, N. (n.d.). Research on Visualisation in Learning and Teaching Sharon McCleary 11
  • 12. Mathematics in Middle and Upper Primary EDUC8505 Mathematics, Illinois State University. Radford, L., Schubring, G. & Seeger, F. (2011). Signifying and meaning-making in mathematical thinking, teaching an learning. DOI: 10.1007/s10649-011-9322-5. Reys, R. & Yang, D.C. (1998). Relationship Between Computational Performance and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan. Journal for Research in Mathematics Education. Vol. 29, No. 2, pg 225-237. Schneider, M., Grabner, R. & Paetsch, J. (2009). Mental Number Line, Number Line Estimation, and Mathematical Achievement: Their Interrelations in Grads 5 and 6. Journal of Educational Psychology. Vol. 101, No. 2. pgs 359- 372. Siegler, R., Thompson, C. & Schneider, M. (2011). An Integrated theory of whole number and fractions development. Cognitive Psychology. Vol. 62. pp273- 296. Smith, C., Solomon, G. & Carey, S. (2005). Never getting to zero: elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology. Vol.51. pp101-140. Stenmark, J. & Bush, W. (2001). Mathematics Assessment, A Practical Handbook. National Council of Teachers of Mathematics. VA. Sullivan, P. & Lilburn, P. (2004). Open-ended Maths Activities, Using ‘good’ questions to enhance learning in Mathematics. 2nd Edition, Oxford University Press, Oxford. The Australian Curriculum-Mathematics, Version 1.1, (2010). Australian Curriculum, Assessment and Reporting Authority [ACARA], Retrieved from: http://www.australiancurriculum.edu.au Vygotsky, L., (1978). Mind in Society, Harvard University Press, Cambridge, MA. Way, J. & Bobis, J. (2011). Fractions, Teaching for Understanding. The Australian Association of Mathematics Teachers Inc. South Australia. Wilkerson-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?: resources and acts for constructing and understanding mathematicians, doi: 10.1007/s10649-011-9306-5. Sharon McCleary 12
  • 13. Mathematics in Middle and Upper Primary EDUC8505 Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K. (2004), First Steps in Mathematics: Number (Book 1). Rigby. Australia. Sharon McCleary 13